3-3 Normal Distribution

MAT2371 Introduction to Probability § 3.3 The Normal Distribution Zao-Li CHEN University of Ottawa Department of Mathematics and Statistics Z.-L. Chen MAT2371B Fall 2023 1 / 14

Why learning normal distributions? Normal distributions are widely seen in real lives. • Errors in measurements, experiments. • Statistics of body weights, heights. • Crop yield. . . . Many quantities follow normal distributions empirically. Z.-L. Chen MAT2371B Fall 2023 2 / 14

Definition Definition 1 ( N ( µ, σ 2 ) ) A r.v. X has a normal distribution if its pdf is f ( x ) = 1 σ √ 2 π exp − ( x − µ ) 2 2 σ 2 , x ∈ R , (1) with parameters µ ∈ R and σ > 0 . Denote by X ∼ N ( µ, σ 2 ) . • µ and σ 2 are, in fact, E ( X ) and Var ( X ) . • N (0 , 1) is called the standard normal distribution . K Write down the pdf for N (0 , 1) . Z.-L. Chen MAT2371B Fall 2023 4 / 14

Properties Let X ∼ N ( µ, σ 2 ) . • The mgf M ( t ) = E ( e tX ) . For all t ∈ R , M ( t ) = Z ∞ −∞ e tx f ( x ) dx = exp µt + σ 2 t 2 2 . (2) K Compute M ′ (0) and M ′′ and show E ( X ) = µ, Var ( X ) = σ 2 . K Let Z = ( X − µ ) /σ . What is E ( Z ) and Var ( Z ) . Theorem 2 The above Z ∼ N (0 , 1) . Z.-L. Chen MAT2371B Fall 2023 5 / 14

Textbook Example 3 . 3-2 Example 4 If the mgf of X is M ( t ) = exp ( 5 t + 12 t 2 ) , t ∈ R . (4) What is the pdf of X ? • Compare (4) with (2). • µ = 5 , σ 2 = 24 , and f ( x ) = 1 √ 48 π exp − ( x − 5) 2 48 , x ∈ R . Z.-L. Chen MAT2371B Fall 2023 7 / 14

Plot a density function • X ∼ N (2 , 9) . ✎ The pdf function, symmetric with respect to the mean. Z.-L. Chen MAT2371B Fall 2023 8 / 14

N(0,1) CDF Take Z ∼ N (0 , 1) , we usually denote by Φ the CDF of Z , Φ( z ) = P { Z ⩽ z } = Z z −∞ ( √ 2 π ) − 1 e − x 2 / 2 dx. (5) Example 5 With the help from textbook Tables V a and V b to compute (1) P {− 1 ⩽ Z ⩽ 1 } . (2) P {− 2 ⩽ Z ⩽ 2 } . (3) P {− 3 ⩽ Z ⩽ 3 } . Z.-L. Chen MAT2371B Fall 2023 10 / 14

N(0,1) CDF Take Z ∼ N (0 , 1) , we usually denote by Φ the CDF of Z , Φ( z ) = P { Z ⩽ z } = Z z −∞ ( √ 2 π ) − 1 e − x 2 / 2 dx. (5) Example 5 With the help from textbook Tables V a and V b to compute (1) P {− 1 ⩽ Z ⩽ 1 } . (2) P {− 2 ⩽ Z ⩽ 2 } . (3) P {− 3 ⩽ Z ⩽ 3 } . (1) P {− 1 ⩽ Z ⩽ 1 } = 1 − P { Z > 1 } − P { Z < − 1 } ≈ 0 . 6826 . (2) P {− 2 ⩽ Z ⩽ 2 } ≈ 0 . 9426 . (3) P {− 3 ⩽ Z ⩽ 3 } ≈ 0 . 9974 . Z.-L. Chen MAT2371B Fall 2023 10 / 14

Textbook Example 3 . 3-7 Example 6 If X ∼ N (25 , 36) , find a constant c > 0 such that P {| X − 25 | < c } = 0 . 9544 . • Z = ( X − 25) / 6 ∼ N (0 , 1) . If suffices find c such that 0 . 9544 = P {− c/ 6 ⩽ Z ⩽ c/ 6 } = Φ( c/ 6) − [1 − Φ( c/ 6)] ⇝ Φ( c/ 6) = 0 . 9772 . • Use the R command, qnorm(0.9772,mean=0, sd=1). c/ 6 ≈ 1 . 9991 , c ≈ 11 . 9946 . Z.-L. Chen MAT2371B Fall 2023 12 / 14

Two Travel Routes Example 7 In a city, two routes to travel from south to north. (1) Through the downtown, shorter distance, heavier traffic, N (50 , 100) minutes. (2) Through the ring road, longer distance, lighter traffic, N (60 , 16) minutes. Suppose you have 80 minutes, which route to choose? Let τ 1 and τ 2 be the travel time respectively. (1) P { τ 1 ⩽ 80 } = P { ( τ − 50) / 10 < (80 − 50) / 10 } = Φ(3) . (2) P { τ 2 ⩽ 80 } = P { ( τ − 60) / 4 < (80 − 60) / 4 } = Φ(5) . Choose the ring road. Z.-L. Chen MAT2371B Fall 2023 13 / 14